 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2008/d/d4/b/ii/solution

It is not Lebesgue integrable, because its absolute integral diverges. On

I_{k} = [kπ + π /6, kπ + 5 π /6]

, one has

∣ sin x ∣ \geq 1/2

and

x \leq (k + 1) π

. Thus

\int_{I_{k}} \frac{∣ s i n x ∣}{x} d x \geq \frac{1}{3 ( k + 1 )} .

These disjoint-interval lower bounds have a divergent harmonic sum. The finite improper value from part (i) is therefore conditional convergence, not Lebesgue integrability.

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